1 Introduction .................................................
2 Continuum model of elastic crystals .......................... 3
2.1 Notation and general background ......................... 3
2.2 Symmetries among the crystallographic parameters for
different twin systems .................................. 6
3 Energy barriers accompanying martensitic nucleation ......... 10
3.1 Incompatibility due to λ2 ≠ 1: Phase mixture ........... 10
3.1.1 Connection to the Kohn-Müller model with
unequal volume fractions ........................ 12
3.2 Incompatibility of average strains: Failure of self-
accommodation .......................................... 19
3.3 Energy barrier and hysteresis .......................... 21
3.3.1 Martensitic nucleus surrounded by austenite ..... 21
3.3.2 Nucleation at the boundary ...................... 24
4 Periodic non-branching transition layers .................... 27
4.1 Influence of reorientation ............................. 27
4.1.1 Piecewise affine transition layers from [131]
revisited ....................................... 27
4.1.2 Symmetric versus asymmetric needles ............. 30
4.2 The minimization problem for the elastic energy ........ 34
4.2.1 Formulation on the reference domain ............. 38
4.2.2 Computation of the Г-limit for λ → 0 ............ 46
5 The Г-limit problem ......................................... 53
5.1 Existence, uniqueness and some regularity of a
solution ............................................... 54
5.2 Needles are pinched .................................... 62
5.2.1 Extensions of fractional order Sobolev
functions ....................................... 62
5.3 Approximate solution of the Г-limit problem ............ 69
5.3.1 Semi-quantitative estimates via Fourier
ansatz .......................................... 70
5.3.2 Numerical approximation via finite elements ..... 97
5.4 Model hierarchy ....................................... 103
6 Multiscale approximation and reproducing kernel Hilbert
space methods .............................................. 110
6.1 Numerical solution to variational inequalities ........ 111
6.2 Notation and basic definitions ........................ 114
6.3 Existence of finite level based approximating
interpolants .......................................... 118
6.4 Bernstein inequalities ................................ 121
6.5 Interpolation with truncated kernel ................... 126
6.6 Derivative-free sampling inequalities ................. 129
A H1/2-seminorm for sawtooth-type functions ................... 132
Bibliography .................................................. 134
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